Question

How do you solve `8x=3x^2 - 1` using completing the square?

Answer 1

`x_1 = 4/3 + sqrt(19)/3`, `x_2 = 4/3 - sqrt(19)/3`

Explanation:

Start by writing your equation in quadratic form

`8x = 3x^2 - 1`

`3x^2 - 8x - 1 = 0`

To solve this equation by completing the square you first need to find a way to write the left side of the equation as a square of a binomial.

Move the term that does not contain `x` or `x^2` to the right side of the equation.

`3x^2 - 8x = 1`

Now divide everything by 3 to get

`(color(red)(cancel(color(black)(3))) * x^2)/color(red)(cancel(color(black)(3))) - 8/3x = 1/3`

`x^2 - 8/3x = 1/3`

Next, divide the coefficient of the `x`-term by 2, square it, then add the result to both sides of the equation. In your case, you have

`(-8/3) * 1/2 = -4/3`, then

`(-4/3)^2 = 16/9`

Your quadratic will become

`x^2 - 8/3x + 16/9 = 1/3 + 16/9`

The left side of the equation can be written as

`(x^2 - 8/3x + 16/9) = (x-4/3)^2`

This will get you

`(x - 4/3)^2 = 19/9`

Take the square roots of both sides to find the two solutions

`sqrt((x-4/3)^2) = sqrt(19/9)`

`x-4/3 = +- sqrt(19)/3 => x_(1,2) = 4/3 +- sqrt(19)/3`

The two solutions will be

`x_1 = color(green)(4/3 + sqrt(19)/3` and `x_2 = color(green)(4/3 - sqrt(19)/3)`